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			<h1 id="firstHeading" class="firstHeading">Sinc filter</h1>
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<div class="thumbinner" style="width:222px;"><a href="http://en.wikipedia.org/wiki/File:Sinc_function_%28normalized%29.svg" class="image"><img alt="" src="wikipedia-Sinc_filter_pliki/220px-Sinc_function_normalized.png" class="thumbimage" height="151" width="220"></a>
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The normalized <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>, the <a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response">impulse response</a> of the sinc filter.</div>
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<div class="thumbinner" style="width:222px;"><a href="http://en.wikipedia.org/wiki/File:Rectangular_function.svg" class="image"><img alt="" src="wikipedia-Sinc_filter_pliki/220px-Rectangular_function.png" class="thumbimage" height="165" width="220"></a>
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The <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rectangular function</a>, the <a href="http://en.wikipedia.org/wiki/Frequency_response" title="Frequency response">frequency response</a> of the sinc filter.</div>
</div>
</div>
<p>In <a href="http://en.wikipedia.org/wiki/Signal_processing" title="Signal processing">signal processing</a>, a <b>sinc filter</b> is an idealized <a href="http://en.wikipedia.org/wiki/Filter_%28signal_processing%29" title="Filter (signal processing)">filter</a> that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has <a href="http://en.wikipedia.org/wiki/Linear_phase" title="Linear phase">linear phase</a>. The filter's <a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response">impulse response</a> is a <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a> in the time domain, and its <a href="http://en.wikipedia.org/wiki/Frequency_response" title="Frequency response">frequency response</a> is a <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rectangular function</a>.</p>
<p>It is an "ideal" <a href="http://en.wikipedia.org/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a>
 in the frequency sense, perfectly passing low frequencies, perfectly 
cutting high frequencies; and thus may be considered to be a <a href="http://en.wikipedia.org/wiki/Brick-wall_filter" title="Brick-wall filter" class="mw-redirect">brick-wall filter</a>.</p>
<p>Real-time filters can only approximate this ideal, since an ideal sinc filter (aka <i>rectangular filter</i>) is <a href="http://en.wikipedia.org/wiki/Causal_filter" title="Causal filter">acausal</a> and has an infinite delay, but it is commonly found in conceptual demonstrations or proofs, such as the <a href="http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">sampling theorem</a> and the <a href="http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">Whittaker–Shannon interpolation formula</a>.</p>
<p>In mathematical terms, the desired frequency response is the <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rectangular function</a>:</p>
<dl>
<dd><img class="tex" alt="H(f) = \mathrm{rect} \left( \frac{f}{2B} \right)" src="wikipedia-Sinc_filter_pliki/0a3acc636058c328226dc327ca8f4ee4.png"></dd>
</dl>
<p>where <img class="tex" alt="B\," src="wikipedia-Sinc_filter_pliki/d5742827d055e40cd568fd71271f4681.png"> is an arbitrary cutoff frequency (aka <i>bandwidth</i>). The impulse response of such a filter is given by the <a href="http://en.wikipedia.org/wiki/Continuous_Fourier_transform#Table_of_important_Fourier_transforms" title="Continuous Fourier transform" class="mw-redirect">inverse Fourier transform</a>:</p>
<dl>
<dd><img class="tex" alt="
\begin{align}
h(t) = \mathcal{F}^{-1} \{ H (f)\} &amp; = 2B \frac{\sin(2\pi Bt)}{2\pi Bt} \\
 &amp; = 2B \, \mathrm{sinc}(2 B t)
\end{align}
" src="wikipedia-Sinc_filter_pliki/38e2f1ab302cb1cfb11e302d709d1cf4.png"></dd>
</dl>
<p>in terms of the normalized <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>.</p>
<p>As the sinc filter has infinite impulse response in both positive and
 negative time directions, it must be approximated for real-world 
(non-abstract) applications; a <a href="http://en.wikipedia.org/wiki/Window_function" title="Window function">windowed</a> sinc filter is often used instead.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Brick-wall_filters"><span class="tocnumber">1</span> <span class="toctext">Brick-wall filters</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Frequency-domain_sinc"><span class="tocnumber">2</span> <span class="toctext">Frequency-domain sinc</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#See_also"><span class="tocnumber">3</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#References"><span class="tocnumber">4</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#External_links"><span class="tocnumber">5</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sinc_filter&amp;action=edit&amp;section=1" title="Edit section: Brick-wall filters">edit</a>]</span> <span class="mw-headline" id="Brick-wall_filters">Brick-wall filters</span></h2>
<p>An idealized <a href="http://en.wikipedia.org/wiki/Electronic_filter" title="Electronic filter">electronic filter</a>,
 one that has full transmission in the pass band, and complete 
attenuation in the stop band, with abrupt transitions, is known 
colloquially as a "brick-wall filter", in reference to the shape of the <a href="http://en.wikipedia.org/wiki/Transfer_function" title="Transfer function">transfer function</a>. The sinc filter is a brick-wall <a href="http://en.wikipedia.org/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a>, from which brick-wall <a href="http://en.wikipedia.org/wiki/Band-pass_filter" title="Band-pass filter">band-pass filters</a> and <a href="http://en.wikipedia.org/wiki/High-pass_filter" title="High-pass filter">high-pass filters</a> are easily constructed.</p>
<p>The lowpass filter with brick-wall cutoff at frequency <i>B</i><sub><i>L</i></sub> has impulse response and transfer function given by:</p>
<dl>
<dd><img class="tex" alt=" h_{LPF}(t) = 2B_L \, \mathrm{sinc}\left(2B_L t\right)" src="wikipedia-Sinc_filter_pliki/7512bdeb1cd8d61dd6e7c0a27d8c43a5.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt=" H_{LPF}(f) = \mathrm{rect}\left( \frac{f}{2B_L} \right)." src="wikipedia-Sinc_filter_pliki/72edcc4dadf8b7fcb19313367947c3d5.png"></dd>
</dl>
<p>The band-pass filter with lower band edge <i>B</i><sub><i>L</i></sub> and upper band edge <i>B</i><sub><i>H</i></sub>
 is just the difference of two such sinc filters (since the filters are 
zero phase, their magnitude responses subtract directly):<sup id="cite_ref-0" class="reference"><a href="#cite_note-0"><span>[</span>1<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt=" h_{BPF}(t) = 2B_H \, \mathrm{sinc}\left(2B_H t\right) - 2B_L \, \mathrm{sinc}\left(2B_L t\right)" src="wikipedia-Sinc_filter_pliki/99bccff3c99d08e491e70405496bc52f.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt=" H_{BPF}(f) = \mathrm{rect}\left( \frac{f}{2B_H} \right) - \mathrm{rect}\left( \frac{f}{2B_L} \right)." src="wikipedia-Sinc_filter_pliki/9a77b296191162fc3515cd2b3ee6879d.png"></dd>
</dl>
<p>The high-pass filter with lower band edge <i>B</i><sub><i>H</i></sub> is just a transparent filter minus a sinc filter, which makes it clear that the <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> is the limit of a narrow-in-time sinc filter:</p>
<dl>
<dd><img class="tex" alt=" h_{HPF}(t) = \delta(t) - 2B_H \, \mathrm{sinc}\left(2B_H t\right)" src="wikipedia-Sinc_filter_pliki/9ec4e61693ce9402a8afd43de229d3fc.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt=" H_{HPF}(f) = 1 - \mathrm{rect}\left( \frac{f}{2B_H} \right)." src="wikipedia-Sinc_filter_pliki/65128cfe3b701ad851612f95a173f46c.png"></dd>
</dl>
<p>Brick-wall filters that run in realtime are not physically realizable as they have infinite latency (i.e., its <a href="http://en.wikipedia.org/wiki/Compact_support" title="Compact support" class="mw-redirect">compact support</a> in the <a href="http://en.wikipedia.org/wiki/Frequency_domain" title="Frequency domain">frequency domain</a>
 forces its time response not to have compact support meaning that it is
 ever-lasting) and infinite order (i.e., the response cannot be 
expressed as a <a href="http://en.wikipedia.org/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equation</a> with a finite sum), but approximate implementations are sometimes used and they are frequently called brick-wall filters.<sup class="Template-Fact" title="This claim needs references to reliable sources from May 2009" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a></i>]</sup></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sinc_filter&amp;action=edit&amp;section=2" title="Edit section: Frequency-domain sinc">edit</a>]</span> <span class="mw-headline" id="Frequency-domain_sinc">Frequency-domain sinc</span></h2>
<p>The name "sinc filter" is applied also to the filter shape that is 
rectangular in time and a sinc function in frequency, as opposed to the 
ideal low-pass sinc filter, which is sinc in time and rectangular in 
frequency. In case of confusion, one may refer to these as <b>sinc-in-frequency</b> and <b>sinc-in-time,</b> according to which domain the filter is sinc in.</p>
<p>Sinc-in-frequency <a href="http://en.wikipedia.org/wiki/Cascaded_integrator-comb" title="Cascaded integrator-comb" class="mw-redirect">CIC</a> filters, among many other applications, are almost universally used for <a href="http://en.wikipedia.org/wiki/Decimation_%28signal_processing%29" title="Decimation (signal processing)">decimating</a> <a href="http://en.wikipedia.org/wiki/Sigma-delta_modulation" title="Sigma-delta modulation" class="mw-redirect">sigma–delta</a> <a href="http://en.wikipedia.org/wiki/Analog-to-digital_converter" title="Analog-to-digital converter">ADCs</a>, as they are easy to implement and nearly optimum for this use.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>2<span>]</span></a></sup></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sinc_filter&amp;action=edit&amp;section=3" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Lanczos_resampling" title="Lanczos resampling">Lanczos resampling</a></li>
<li><a href="http://en.wikipedia.org/wiki/Aliasing" title="Aliasing">Aliasing</a></li>
<li><a href="http://en.wikipedia.org/wiki/Anti-aliasing" title="Anti-aliasing" class="mw-redirect">Anti-aliasing</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sinc_filter&amp;action=edit&amp;section=4" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<div class="reflist" style="list-style-type: decimal;">
<ol class="references">
<li id="cite_note-0"><b><a href="#cite_ref-0">^</a></b> <span class="citation book">Mark Owen (2007). <a href="http://books.google.com/?id=lx-tqq-MkK0C&amp;pg=RA1-PA81&amp;dq=sinc-function+high-pass+band-pass+difference&amp;q=sinc-function%20high-pass%20band-pass%20difference" class="external text" rel="nofollow"><i>Practical signal processing</i></a>. Cambridge University Press. p.&nbsp;81. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/9780521854788" title="Special:BookSources/9780521854788">9780521854788</a><span class="printonly">. <a href="http://books.google.com/?id=lx-tqq-MkK0C&amp;pg=RA1-PA81&amp;dq=sinc-function+high-pass+band-pass+difference&amp;q=sinc-function%20high-pass%20band-pass%20difference" class="external free" rel="nofollow">http://books.google.com/?id=lx-tqq-MkK0C&amp;pg=RA1-PA81&amp;dq=sinc-function+high-pass+band-pass+difference&amp;q=sinc-function%20high-pass%20band-pass%20difference</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Practical+signal+processing&amp;rft.aulast=Mark+Owen&amp;rft.au=Mark+Owen&amp;rft.date=2007&amp;rft.pages=p.%26nbsp%3B81&amp;rft.pub=Cambridge+University+Press&amp;rft.isbn=9780521854788&amp;rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3Dlx-tqq-MkK0C%26pg%3DRA1-PA81%26dq%3Dsinc-function%2Bhigh-pass%2Bband-pass%2Bdifference%26q%3Dsinc-function%2520high-pass%2520band-pass%2520difference&amp;rfr_id=info:sid/en.wikipedia.org:Sinc_filter"><span style="display: none;">&nbsp;</span></span></li>
<li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="citation Journal">Chou, W.; Meng, T.H.; Gray, R.M. (1990). "Time domain analysis of sigma delta modulation". <i>Acoustics, Speech, and Signal Processing</i> <b>3</b>: 1751–1754. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a href="http://dx.doi.org/10.1109%2FICASSP.1990.115820" class="external text" rel="nofollow">10.1109/ICASSP.1990.115820</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Time+domain+analysis+of+sigma+delta+modulation&amp;rft.jtitle=Acoustics%2C+Speech%2C+and+Signal+Processing&amp;rft.aulast=Chou%2C+W.%3B+Meng%2C+T.H.%3B+Gray%2C+R.M.&amp;rft.au=Chou%2C+W.%3B+Meng%2C+T.H.%3B+Gray%2C+R.M.&amp;rft.date=1990&amp;rft.volume=3&amp;rft.pages=1751%E2%80%931754&amp;rft_id=info:doi/10.1109%2FICASSP.1990.115820&amp;rfr_id=info:sid/en.wikipedia.org:Sinc_filter"><span style="display: none;">&nbsp;</span></span></li>
</ol>
</div>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sinc_filter&amp;action=edit&amp;section=5" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><a href="http://www.audioholics.com/education/audio-formats-technology/brick-wall-digital-filters-and-phase-deviations" class="external text" rel="nofollow">Brick Wall Digital Filters and Phase Deviations</a></li>
<li><a href="http://www.sweetwater.com/expert-center/glossary/t--BrickwallFilter" class="external text" rel="nofollow">Brick-wall filters</a></li>
</ul>


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